Since the discovery of the fractional quantum Hall effect in 1982, topological phases of electrons have been a subject of great interest. Many abelian topological phases have been discovered in the context of the quantum Hall regime. More recently, high-temperature superconductivity and other complex materials have provided the impetus for further theoretical studies of and experimental searches for abelian topological phases. The types of microscopic models admitting such phases are now better understood. Much less is known about non-abelian topological phases. They are reputed to be obscure and complicated, and there has been little experimental motivation to consider non-abelian topological phases. However, non-abelian topological states would be an attractive milieu for quantum computation.
It has become increasingly clear that if a new generation of computers could be built to exploit quantum mechanical superpositions, enormous technological implications would follow. In particular, solid state physics, chemistry, and medicine would have a powerful new tool, and cryptography also would be revolutionized.
The standard approach to quantum computation is predicated on the quantum bit (“qubit”) model in which one anticipates computing on a local degree of freedom such as a nuclear spin. In a qubit computer, each bit of information is typically encoded in the state of a single particle, such as an electron or photon. This makes the information vulnerable. If a disturbance in the environment changes the state of the particle, the information is lost forever. This is known as decoherence—the loss of the quantum character of the state (i.e., the tendency of the system to become classical). All schemes for controlling decoherence must reach a very demanding and possibly unrealizable accuracy threshold to function.
Topology has been suggested to stabilize quantum information. A topological quantum computer would encode information not in the conventional zeros and ones, but in the configurations of different braids, which are similar to knots but consist of several different threads intertwined around each other. The computer would physically weave braids in space-time, and then nature would take over, carrying out complex calculations very quickly. By encoding information in braids instead of single particles, a topological quantum computer does not require the strenuous isolation of the qubit model and represents a new approach to the problem of decoherence.
In 1997, there were independent proposals by Kitaev and Freedman that quantum computing might be accomplished if the “physical Hilbert space” V of a sufficiently rich TQFT (topological quantum field theory) could be manufactured and manipulated. Hilbert space describes the degrees of freedom in a system. The mathematical construct V would need to be realized as a new and remarkable state for matter and then manipulated at will.
In 2000, Freedman showed that some extraordinarily complicated local Hamiltonian H can be written down whose ground state is V. But this H is an existence theorem only, and is far too complicated to be the starting point for a physical realization.
In 2002, Freedman showed a Hamiltonian involving four-body interactions and stated that after a suitable perturbation, the ground state manifold of H will be the desired state V. This H is less complex than the previously developed H, but it is still only a mathematical construct. One does not see particles, ions, electrons, or any of the prosaic ingredients of the physical world in this prior art model. A Hamiltonian is an energy operator that describes all the possible physical states (eigenstates) of the system and their energy values (eigenvalues).
Freedman further defined the notion of d-isotopy, and showed that if it can be implemented as a ground state of a reasonable Hamiltonian, then this would lead to V and to topological quantum computation. Isotopy is defined as deformation, and two structures that are isotopic are considered to be the same. As shown in the toruses 1 and 2 of FIGS. 1A and 1B, respectively, for example, X and X′ are isotopic, because one may be gradually deformed into the other. In d-isotopy, small circles can be absorbed as a factor=d. Such closed curves are referred to as multicurves or multiloops. Loop X″ in FIG. 1C (winding around torus 3) is not d-isotopic to X or X′. Loops that are unimportant (because, e.g., they comprise a contractible circle) are called trivial loops and it is desirable to remove, as well as count them. Whenever a trivial loop is removed, the picture is multiplied by “d”. In other words, if two multiloops are identical except for the presence of a contractible circle, then their function values differ by a factor of d, a fixed positive real number. It has been shown that d=2 cos π/(k+2), where k is a level such as 1, 2, 3, etc. which is a natural parameter of Cherns-Simons theory.
According to Freedman, the parameter d can take on only the “special” values: 1, root2, golden ratio, root3 . . . 2 cos π/(k+2) (where k is a natural number). At d=1, the space V becomes something already known, if not observed in solid state physics. For d>1, V is new to the subject. Freedman, et al., later showed that d-isotopy is explicable by field theory and that multiloops as domain walls can be alternately interpreted as Wilson loop operators. Thus, d-isotopy is a mathematical structure that can be imposed on the multiloops, and is based on Cherns-Simons theory.
An exotic form of matter is a fractional quantum Hall fluid. It arises when electrons at the flat interface of two semiconductors are subjected to a powerful magnetic field and cooled to temperatures close to absolute zero. The electrons on the flat surface form a disorganized liquid sea of electrons, and if some extra electrons are added, quasi-particles called anyons emerge. Unlike electrons or protons, anyons can have a charge that is a fraction of a whole number.
The fractional quantum Hall fluids at one-third filling (of the first Landau level) are already a rudimentary (abelian) example of the V of a TQFT. To effect quantum computation, it would be desirable to construct states more stable and more easily manipulated than FQHE (fractional quantum Hall effect) fluids.
One property of anyons is that when they are moved around each other, they remember in a physical sense the knottedness of the paths they followed, regardless of the path's complexity. It is desirable to use anyons in a system with complex enough transformations, called non-abelian transformations, to carry out calculations in a topological quantum computation system.
In view of the foregoing, there is a need for systems and methods that overcome the limitations and drawbacks of the prior art.